Theorem cbv3hv 1850

Description: Lemma for ax10 1944. Similar to cbv3h 1983. Requires distinct variables but avoids ax-12 1925. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Hypotheses

Ref	Expression
cbv3hv.1	|- (ph -> A.yph)
cbv3hv.2	|- (ps -> A.xps)
cbv3hv.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbv3hv	|- (A.xph -> A.yps)

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)   ps(x,y)

Proof of Theorem cbv3hv

Step	Hyp	Ref	Expression
1		cbv3hv.1	. . 3 |- (ph -> A.yph)
2	1	alimi 1559	. 2 |- (A.xph -> A.xA.yph)
3		a9ev 1656	. . . . . . 7 |- E.x x = y
4		cbv3hv.3	. . . . . . . 8 |- (x = y -> (ph -> ps))
5	4	eximi 1576	. . . . . . 7 |- (E.x x = y -> E.x(ph -> ps))
6	3, 5	ax-mp 5	. . . . . 6 |- E.x(ph -> ps)
7	6	19.35i 1601	. . . . 5 |- (A.xph -> E.xps)
8		cbv3hv.2	. . . . . 6 |- (ps -> A.xps)
9	8	19.9h 1780	. . . . 5 |- (E.xps <-> ps)
10	7, 9	sylib 188	. . . 4 |- (A.xph -> ps)
11	10	alimi 1559	. . 3 |- (A.yA.xph -> A.yps)
12	11	a7s 1735	. 2 |- (A.xA.yph -> A.yps)
13	2, 12	syl 15	1 |- (A.xph -> A.yps)

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)